A PROOF OF THE INVARIANT SUBSPACE CONJECTURE Louis de Branges

A PROOF OF THE INVARIANT SUBSPACE CONJECTURE

Louis de Branges∗

Abstract. A proof is given of the announced existence theorem for invariant subspaces of

continuous (linear) transformations of a Hilbert space into itself [5]. If the transformation isnot a scalar multiple of the identity transformation, a closed subspace other than the least

subspace and the greatest subspace exists which is an invariant subspace for every continuous

linear transformation which commutes with the given transformation.

Since a continuous transformation of a Hilbert space into itself is a scalar multiple of acontractive transformation, it is sufficient to prove the existence of invariant subspaces fortransformations which are contractive. The existence of invariant subspaces is a theoremof Hilbert for transformations which map a Hilbert space isometrically onto itself. Whena contractive transformation A maps a Hilbert space into itself, the set of element f of thespace which have the same norm as Anf for every nonnegative integer n is an invariantsubspace for every continuous transformation which commutes with A. The existence ofthe desired invariant subspace follows when the subspace contains a nonzero element. Theexistence of invariant subspaces is now obtained when the subspace contains no nonzeroelement. A canonical model of the transformation is applied. A contractive transformationC onto a coefficient Hilbert C exists such that the identity

‖Af‖2H = ‖f‖2H − |Cf |2

holds for every element f of the Hilbert space H. The absolute value denotes the norm inthe coefficient Hilbert space. An element f of H is uniquely determined by the associatedpower series

Cf + CAfz + CA2fz2 + . . .

whose coefficients are taken in the coefficient space. The given transformation is unitarilyequivalent to the transformation which takes f(z) into [f(z)− f(0)]/z in a Hilbert spacewhose elements are power series with coefficients in C. The existence of invariant subspacesis proved for transformations given in this canonical form.

∗ Research supported by the National Science Foundation

Mathematical Reviews Classification 47A15

1

2 LOUIS DE BRANGES December 23, 2008

In the construction of invariant subspaces a vector is an element of the coefficient space.An operator is a continuous transformation of vectors into vectors. If b is a vector, b− isthe linear functional on vectors defined by the scalar product

b−a = 〈a, b〉

in the coefficient space. If a and b are vectors, the operator ab− is defined by the associativelaw

(ab−)c = a(b−c)

for every vector c. Complex numbers are treated as multiplication operators. The absolutevalue denotes the operator norm of an operator as well as the norm of a vector. A bardenotes the adjoint A− of an operator A.

A Hilbert space H, whose elements are power series with vector coefficients, is saidto satisfy the inequality for difference quotients if [f(z) − f(0)]/z belongs to the spacewhenever f(z) belongs to the space and if the inequality for difference quotients

‖[f(z)− f(0)]/z‖2H ≤ ‖f(z)‖2H − |f(0)|2

is satisfied. The space is said to satisfy the identity for difference quotients if equalityalways holds in the inequality for difference quotients.

The space C(z) of square summable power series is a Hilbert space of power series withvector coefficients which satisfies the identity for difference quotients. The elements of thespace are the power series

f(z) = a0 + a1z + a2z2 + . . .

with vector coefficient such that the sum

‖f(z)‖2 = |a0|2 + |a1|2 + |a2|2 + . . .

is finite. A Hilbert space H of power series with vector coefficients which satisfies theinequality for difference quotients is contained contractively in C(z). The spaces whichare contained isometrically in C(z) are treated by Beurling [1] in the essential case ofthe complex numbers as coefficient space. A generalization of the Beurling theory tocontractive inclusions is made in joint work with James Rovnyak [6].

The treatment of contractive inclusions applies a generalization of orthogonal comple-mentation [3]. If a Hilbert space P is contained contractively in a Hilbert space H, a uniqueHilbert space Q, which is contained contractively in H, exists such that the inequality

‖c‖2H ≤ ‖a‖2P + ‖b‖2Q

holds wheneverc = a+ b

A PROOF OF THE INVARIANT SUBSPACE CONJECTURE 3

is the sum of an element a of P and an element b of Q and such that every element c of Hadmits a decomposition for which equality holds. The space Q is called the complementaryspace to P in H. The minimal decomposition of an element c of H is unique. The elementa of P is obtained from c under the adjoint of the inclusion of P in H. The element b ofQ is obtained from c under the adjoint of the inclusion of Q in H. The intersection of Pand Q is a Hilbert space P ∧Q, which is contained contractively in H, with scalar productdetermined by the identity

‖c‖2P∧Q = ‖c‖2P + ‖c‖2Q.

The Hilbert space H in which P and Q are contained contractively as complementaryspaces is denoted P ∨Q. The inequality

‖c‖2P∨Q ≤ ‖a‖2P + ‖b‖2Q

holds whenever

c = a+ b

is the sum of an element a of P and an element b of Q. Every element c of P ∨Q admitsa unique decomposition for which equality holds.

Complementation is preserved under partially isometric transformations. If Hilbertspaces P and Q are contained contractively as complementary subspaces of a Hilbert spaceP ∨ Q and if T is a partially isometric transformation of P ∨ Q onto a Hilbert space H,then Hilbert spaces P ′ and Q′ exists which are contained contractively as complementarysubspaces of the space

H = P ′ ∨Q′

such that T acts as a partially isometric transformation of P onto P ′ and of Q onto Q′.If a Hilbert space H of power series with vector coefficients satisfies the inequality for

difference quotients, then the space is contained contractively in the space C(z) of squaresummable power series. The complementary space to H in C(z) is a Hilbert space M,which is contained contractively in C(z), such that multiplication by z is a contractivetransformation of M into itself. If a given Hilbert space M is contained contractively inC(z) and if multiplication by z is a contractive transformation of M into itself, then thecomplementary space to M in C(z) is a Hilbert space H which satisfies the inequality fordifference quotients.

If multiplication by a power series W (z) with operator coefficients is a contractivetransformation of C(z) into itself, then multiplication by W (z) acts as a partially isometrictransformation of C(z) onto a Hilbert spaceM such that multiplication by z is a contractivetransformation ofM into itself. The complementary space H(W ) toM in C(z) is a Hilbertspace of power series with vector coefficients which satisfies the inequality for differencequotients.

If a Hilbert space H of power series with vector coefficients satisfies the inequality fordifference quotients, an augmented space H′ of power series with vector coefficients isconstructed which satisfies the inequality for difference quotients. The elements of H′ are


the power series f(z) with vector coefficients such that [f(z)− f(0)]/z belongs to H. Thescalar product in H′ is determined by the identity for difference quotients

‖[f(z)− f(0)]/z‖2H = ‖f(z)‖2H′ − |f(0)|2.

The given space H is contained contractively in the augmented space H′. If the dimensionof the complementary space to H in H′ is less than or equal to the dimension of thecoefficient space, a partially isometric transformation exists of the coefficient space C ontothe complementary space. A power series W (z) with operator coefficients exists suchthat multiplication by W (z) acts as a partially isometric transformation of C onto thecomplementary space toH inH′. Multiplication by W (z) is a contractive transformation ofC(z) into itself. The given space H is isometrically equal to the complementary spaceH(W )to the range of multiplication by W (z) in C(z). The augmented space H′ is isometricallyequal to the complementary space to the range of multiplication by zW (z) in C(z).

The construction of a space H(W ) is possible when a Hilbert space M is containedcontractively in C(z) and when multiplication by z is an isometric transformation of thespace M into itself. The complementary space H to M in C(z) then satisfies the inequal-ity for difference quotients. The space H is isometrically equal to a space H(W ) if thedimension of the complementary space to H in its augmented space H′ is less than or equalto the dimension of the coefficient space C. The complementary space to H′ in C(z) is aHilbert space M′, which is contained contractively in M, such that multiplication by zis an isometric transformation of M onto M′. The complementary space to H in H′ isisometrically equal to the complementary space to M′ in M.

The dimension estimate is satisfied when the space H satisfies the identity for differencequotients.

Theorem 1. A Hilbert space H of power series with vector coefficients which satisfiesthe identity for difference quotients is isometrically equal to a space H(W ) for a powerseries W (z) with operator coefficients such that multiplication by W (z) is a contractivetransformation of C(z) into itself whose kernel contains [f(z)−f(0)]/z whenever it containsf(z).

The theorem is due to Beurling [1] for a space H which is contained isometrically inC(z) in the essential case of the complex numbers as coefficient space. The proof of thetheorem is prepared by a dimension estimate obtained in joint work with James Rovnyakas a undergraduate at Lafayette College and as a graduate student of Yale Universitybefore he came to Purdue University in 1963 in a postdoctoral position. A Hilbert spacewhich is contained contractively in C(z) and satisfies the inequality for difference quotientshas a complementary space M in C(z) such that multiplication by z is a contractivetransformation of M into itself.

Lemma. If a Hilbert space M is contained contractively in C(z) and if multiplication byz is an isometric transformation of M into itself, then the dimension of the orthogonalcomplement inM of the range of multiplication by z is less than or equal to the dimensionof the coefficient space.


Proof of Lemma. The dimension estimate is satisfied when the coefficient space has infinitedimension since the dimension of M is less than or equal to the dimension of C(z) whichis equal to the dimension of C. It remains to obtain the dimension estimate when thecoefficient space has finite dimension r for some positive integer r.

Argue by contradiction assuming that the orthogonal complement ofM′ inM containsr + 1 linearly independent elements

f0(z), . . . , fr(z).

If c0, . . . , cr are corresponding vectors, then the square matrix which has entry

c−i fj(z)

in the i–th row and j–th column has vanishing determinant. The determinant is a powerseries with complex coefficients which represents a bounded function in the unit disk.Expansion of the determinant on the row i equal to zero produces the vanishing powerseries

c−0 f0(z)g0(z) + . . .+ c−0 fr(z)gr(z)

with(−1)kgk(z)

the determinant of the matrix obtained by deleting the 0–th row and k–th column fromthe starting matrix. Since the vector c0 is arbitrary, the element

f0(z)g0(z) + . . .+ fr(z)gr(z)

of M vanishes identically. Each power series

gk(z) = ak0 + ak1z + ak2z2 + . . .

with complex coefficients is square summable since it represents a bounded function in theunit disk. Since multiplication by z is isometric in M and since

f0(z), . . . , fr(z)

are orthogonal to the range of the transformation, the elements

zmf0(z), . . . , zmfr(z)

of M are orthogonal to the elements

znf0(z), . . . , znfr(z)

when m and n are distinct nonnegative integers. The element

f0(z)a0nzn + . . .+ fr(z)arnz

n


vanishes for every nonnegative integer n. Since the elements

f0(z), . . . , fr(z)

of M are linearly independent, all coefficients of the power series

g0(z), . . . , gr(z)

vanish identically.

Since the power series are determinants of square matrices of the same form as thestarting matrix, the construction can be repeated to produce square matrices of smallerorder with vanishing determinant. A contradiction is obtained since all coefficients

c−i fj(z)

of the starting matrix then vanish identically.

This completes the proof of the theorem.

The space H(W ) is characterized as the state space of a linear system. The maintransformation A, which takes f(z) into [f(z)− f(0)]/z, maps the state space into itself.The input transformation B, which takes c into [W (z) −W (0)]c/z, maps the coefficientspace into the state space. The output transformation C, which takes f(z) into f(0),maps the state space into the coefficient space. The external operator D, which takes cinto W (0)c, maps the coefficient space into itself. The matrix of the linear system(

A BC D

)maps the Cartesian product of the state space and the coefficient space into itself. Thelinear system is coisometric since the matrix has an isometric adjoint. The linear systemis canonical since a canonical model of the state space is applied. The transfer functionW (z) of the linear system is a Schur function since it is characterized as a power serieswith operator coefficients which represents a function with contractive values in the unitdisk.

A convex structure applies to the Hilbert spaces which are contained contractively in aHilbert space H. If Hilbert spaces P and Q are contained contractively in H and if t is inthe interval [0, 1], a unique Hilbert space

S = (1− t)P + tQ

exists, which is contained contractively in H, such that the convex combination

c = (1− t)a+ tb

belongs to S whenever a belongs to P and b belongs to Q, such that the inequality

‖c‖2S ≤ (1− t)‖a‖2P + t‖b‖2Q


is satisfied, and such that every element c of S is a convex combination for which equalityholds. The adjoint of the inclusion of S in H is the convex combination

(1− t)P + tQ

of the adjoint P of the inclusion of P in H and the adjoint Q of the inclusion of Q in H.

A topology applies to the convex set of Hilbert spaces which are contained contractivelyin a Hilbert space H. For the definition of a topology a space is identified with thecontractive transformation of H into itself which extends the adjoint of the inclusion of thespace in H. The space of continuous transformations of H into itself is a Hausdorff space inthe weak topology induced by duality with the space of continuous transformations whichare of trace class. The convex set of contractive transformations of H into itself whichare nonnegative is compact in the subspace topology. If a Hilbert space P is containedcontractively in H, then the adjoint of the inclusion of P in H agrees with a nonnegativeand contractive transformation P . If P is a nonnegative and contractive transformation,a unique Hilbert space P exists which is contained contractively in H and which has Pas the adjoint of its inclusion in H. The convex set of Hilbert spaces which are containedcontractively in H is a compact Hausdorff space in the topology of associated nonnegativeand contractive transformations.

A convex combination of Hilbert spaces of power series with vector coefficients whichsatisfy the inequality for difference quotients is a Hilbert space of power series with vectorcoefficients which satisfies the inequality for difference quotients. The convex set of Hilbertspaces of power series with vector coefficients which satisfy the inequality for differencequotients is a closed subset of the convex set of all Hilbert spaces which are contained con-tractively in C(z). The convex set of Hilbert spaces of power series with vector coefficientswhich satisfy the inequality for difference quotients is the closed convex span of its extremepoints by the Krein–Milman theorem.

A Hilbert space H of power series with vector coefficients which satisfies the identity fordifference quotients is an extreme point of the convex set of Hilbert spaces of power serieswith vector coefficients which satisfy the inequality for difference quotients. If

H = (1− t)P + tQ

is a convex combination with t in the interval (0, 1) of Hilbert spaces P and Q of powerseries with vector coefficients which satisfy the inequality for difference quotients, thenevery element

h(z) = (1− t)f(z) + tg(z)

of H is a convex combination of elements f(z) of P and g(z) of Q such that equality holdsin the inequality

‖h(z)‖2H ≤ (1− t)‖f(z)‖2P + t‖g(z)‖2Q.

Since the inequality

‖[h(z)− h(0)]/z‖2H ≤ (1− t)‖[f(z)− f(0)]/z‖2P + t‖[g(z)− g(0)]/z‖2Q


holds with

‖[f(z)− f(0)]/z‖2P ≤ ‖f(z)‖2P − |f(0)|2

and

‖[g(z)− g(0)]/z‖2Q ≤ ‖g(z)‖2Q − |g(0)|2,

the inequality

‖[h(z)− h(0)]/z‖2H ≤ ‖h(z)‖2H − (1− t)|f(0)|2 − t|g(0)|2

is satisfied. The inequality reads

‖[h(z)− h(0)]/z‖2H ≤ ‖h(z)‖2H − |h(0)|2 − t(1− t)|g(0)− f(0)|2

since the convexity identity

|(1− t)f(0) + tg(0)|2 + t(1− t)|g(0)− f(0)|2 = (1− t)|f(0)|2 + t|g(0)|2

is satisfied. Since the space H satisfies the identity for difference quotients, the constantcoefficients of f(z), g(z), and h(z) are equal. An inductive argument shows that the n-thcoefficients of f(z), g(z), and h(z) are equal for every nonnegative integer n. Since f(z),g(z), and h(z) are always equal, the spaces P, Q, and H are isometrically equal.

The construction of invariant subspaces applies Hilbert spaces of power series with vectorcoefficients which originate in the construction by David Hilbert of invariant subspaces foran isometric transformation of a Hilbert space onto itself. An approach to the spectraltheory of unitary transformations which is close to the methods of Hilbert is preservedin perturbation theory. A canonical model of pairs of unitary transformations acting inthe same Hilbert space is applied which is related to the canonical model of contractivetransformations. Lawrence Shulman in his thesis [8] applies the model in a computationof scattering operators showing unitary equivalence of components of the spectrum. Thedomain and range of scattering operators are conjectured to be determined by squaresummability of power series appearing in the canonical model [7].

A Herglotz space is a Hilbert space, whose elements are power series with vector coeffi-cients, such that a contractive transformation of the space into itself is defined by takingf(z) into [f(z)−f(0)]/z, such that the adjoint transformation is isometric, and such that acontinuous transformation of the space into the coefficient space is defined by taking f(z)into f(0). If H is a Herglotz space, a power series ϕ(z) with operator coefficients, which isunique within an added skew–conjugate operator, exists such that the series

12 [ϕ(z) + ϕ(0)−]c

belongs to the space for every vector c and such that the identity

c−f(0) = 〈f(z), 12 [ϕ(z) + ϕ(0)−]c〉H


holds for every element f(z) of the space. The elements of a Herglotz space are powerseries which converge in the unit disk. If f(z) is an element of a Herglotz space, elementsfn(z) of the space are defined inductively for nonnegative integers n by

f0(z) = f(z)

and byfn+1(z) = [fn(z)− fn(0)]/z

for every nonnegative integer n. The series

[zf(z)− wf(w)]/(z − w) = f0(z) + wf1(z) + w2f2(z) + . . .

converges in the metric topology of the space when w is in the unit disk. A continuoustransformation of the space into the coefficient space is defined by taking a power seriesf(z) into the value f(w) at w of the represented function when w is in the unit disk. Thepower series ϕ(z) converges in the unit disk and represents a function with value ϕ(w) atw. The series

12 [ϕ(z) + ϕ(w)−]c/(1− w−z)

belongs to the Herglotz space for every vector c when w is in the unit disk. The identity

c−f(w) = 〈f(z), 12[ϕ(z) + ϕ(w)−]c/(1− w−z)〉H

holds for every element f(z) of the space. The Herglotz spaceH determined by the functionφ(z) is denoted L(φ).

The space C(z) of square summable power series is isometrically equal to a Herglotzspace L(φ) with

φ(z) = 1.

A Herglotz space L(φ) is contained contractively in a Herglotz space L(ψ) if, and only if,a Herglotz space L(θ) exists with

θ(z) = ψ(z)− φ(z).

The Herglotz space L(θ) is then isometrically equal to the complementary space to thespace L(φ) in the space L(ψ). A convex combination

(1− t)L(φ) + tL(θ)

of Herglotz spaces L(φ) and L(θ) is a Herglotz space L(ψ) with

ψ(z) = (1− t)φ(z) + tθ(z).

The Herglotz spaces which are contained contractively in C(z) form a compact convexset. The convex set is the closed convex span of its extreme points by the Krein–Milmantheorem.


A Herglotz space is associated with a power series W (z) with operator coefficientssuch that multiplication by W (z) is a contractive transformation of C(z) into itself. Theadjoint of multiplication by W (z) is a contractive transformation of C(z) into itself whichtakes [f(z)− f(0)]/z into [g(z)− g(0)]/z whenever it takes f(z) into g(z). The adjoint ofmultiplication by W (z) acts as a partially isometric transformation of C(z) onto a Herglotzspace L(φ) which is contained contractively in C(z). The complementary space in C(z) tothe Herglotz space L(φ) is the Herglotz space L(1− φ).

An element f(z) of C(z) belongs to the space L(1−φ) if, and only if, W (z)f(z) belongsto the space H(W ). The identity

‖f(z)‖2L(1−φ) = ‖f(z)‖2 + ‖W (z)f(z)‖2H(W )

holds for every element f(z) of the space L(1 − φ). An element f(z) of C(z) belongs tothe space H(W ) if, and only if, the adjoint of multiplication by W (z) maps f(z) into anelement g(z) of the space L(1− φ). The identity

‖f(z)‖2H(W ) = ‖f(z)‖2 + ‖g(z)‖2L(1−φ)

holds for every element f(z) of the space H(W ).

The Herglotz space L(1−φ) associated with a space H(W ) was introduced in a graduatecourse taught at Purdue University in the fall semester 1963. A fundamental property ofthe space was discovered by Virginia Rovnyak as a graduate student in the course. Thespace H(W ) satisfies the identity for difference quotients if, and only if, the inclusion ofthe Herglotz space in C(z) is isometric on polynomial elements of the space. The presentconstruction of the space applies the Lowdenslager condition for Wiener factorization [9].

Proof of Theorem 1. A Hilbert space H of power series with vector coefficients which satis-fies the inequality for difference quotients is contained contractively in C(z). Multiplicationby z is a contractive transformation of the complementary space M to H in C(z) into M.The space H is shown isometrically equal to a space H(W ) by showing that multiplicationby z is an isometric transformation of M into itself when H satisfies the identity for dif-ference quotients. The existence of the power series W (z) is then given by the dimensionestimate of the lemma.

It is sufficient to obtain the isometric property of multiplication by z in M when thespace H is known to be a space H(W ) since the coefficient space can be enlarged. Multipli-cation by W (z) is a contractive transformation of C(z) into itself which acts as a partiallyisometric transformation of C(z) ontoM. The isometric property of multiplication by z inM is proved by showing that the kernel of multiplication by W (z) contains [f(z)−f(0)]/zwhenever it contains f(z).

The adjoint of multiplication by W (z) as a transformation of C(z) into itself acts as apartially isometric transformation of C(z) onto a Herglotz space L(φ) which is containedcontractively in C(z) and whose complementary space in C(z) is a Herglotz space L(1−φ).

The space H(W ) is contained contractively in the space H(W ′),

W ′(z) = zW (z).


Multiplication by W ′(z) is a contractive transformation of C(z) into itself which is thecomposition of the contractive multiplication by W (z) and the isometric multiplication byz. The composition can be taken in either order. The adjoint of multiplication by W ′(z)acts as a partially isometric transformation of C(z) onto a Herglotz space L(φ′) which iscontained contractively in C(z) and whose complementary space in C(z) is a Herglotz spaceL(1− φ′).

The adjoint of multiplication by z is a partially isometric transformation of C(z) intoitself which takes f(z) into [f(z) − f(0)]/z. A partially isometric transformation of theHerglotz space L(φ) onto the Herglotz space L(φ′) is defined by taking f(z) into [f(z) −f(0)]/z. A partially isometric transformation of the space L(1−φ) onto the Herglotz spaceL(1− φ′) is defined by taking f(z) into [f(z)− f(0)]/z.

The adjoint of multiplication by W (z) as a transformation of C(z) into itself takes anelement f(z) of the space H(W ) into an element g(z) of the space L(1−φ) which satisfiesthe identity

‖f(z)‖2H(W ) = ‖f(z)‖2 + ‖g(z)‖2L(1−φ).

The adjoint of multiplication by W ′(z) as a transformation of C(z) into itself takes f(z)into the element [g(z)− g(0)]/z of the space L(1− φ′) which satisfies the identity

‖f(z)‖2H(W ′) = ‖f(z)‖2 + ‖[g(z)− g(0)]/z‖2L(1−φ′).

The inclusion of the space H(W ) in the space H(W ′) is isometric on f(z) if, and only if,g(z) is orthogonal to elements of the coefficient space which belong to L(1− φ).

The adjoint of multiplication by W (z) as a transformation of the Herglotz space L(1−φ) into the space H(W ) is the restriction of the adjoint of multiplication by W (z) as atransformation of C(z) into itself.

The kernel of multiplication by W (z) is the orthogonal complement in the Herglotz spaceL(1−φ) of the elements obtained from the space H(W ) under the adjoint of multiplicationby W (z). Since the identity for difference quotients holds in the space H(W ) if, and only if,the space H(W ) is contained isometrically in the space H(W ′), the identity for differencequotients holds in the space H(W ) if, and only if, the kernel of multiplication by W (z)contains the elements of the coefficient space which belong to the space L(1− φ).

Since the set of elements of the Herglotz space L(1−φ) which are obtained from the spaceH(W ) under the adjoint of multiplication by W (z) contains [f(z) − f(0)]/z whenever itcontains f(z), the kernel of multiplication by W (z) is an invariant subspace for the adjointof the transformation taking f(z) into [f(z)−f(0)]/z of the Herglotz space into itself. Sincethe inclusion of the Herglotz space in C(z) is isometric on the kernel of multiplication byW (z), the adjoint agrees with multiplication by z on the kernel of multiplication by W (z).

The orthogonal complement of the kernel of multiplication byW (z) in the Herglotz spaceL(1− φ) is an invariant subspace for the transformation taking f(z) into [f(z)− f(0)]/zon which the transformation is isometric when the space H(W ) satisfies the identity fordifference quotients. The inverse transformation is an isometric transformation of theorthogonal complement of the kernel onto itself. Since the orthogonal complement of


the kernel is an invariant subspace for the adjoint of the transformation taking f(z) into[f(z)− f(0)]/z of the Herglotz space into itself, the kernel of multiplication by W (z) is aninvariant subspace for the transformation.


If a Hilbert space H of power series with vector coefficients satisfies the inequality fordifference quotients, a greatest subspace of H exists which is contained isometrically in Hand which satisfies the identity for difference quotients. An element f(z) of H belongs tothe subspace if the elements fn(z) of H defined inductively by

f0(z) = f(z)

andfn+1(z) = [fn(z)− fn(0)]/z

satisfy the identity for difference quotients

‖fn+1(z)‖2H = ‖fn(z)‖2H − |fn(0)|2

for every nonnegative integer n. The subspace is isometrically equal to a space H(U)for a power series U(z) with operator coefficients such that multiplication by U(z) is acontractive transformation of C(z) into itself whose kernel contains [f(z)−f(0)]/z wheneverit contains f(z).

An element of the orthogonal complement of H(U) in H is a product

U(z)f(z)

with f(z) an element of C(z) which is orthogonal to the kernel of multiplication by U(z).A Hilbert space H′ exists which satisfies the inequality for difference quotients and whichis orthogonal in C(z) to the kernel of multiplication by U(z) such that multiplication byU(z) acts as an isometric transformation of H′ onto the orthogonal complement of H(U) inH. There is no nonzero element in the greatest subspace of H′ which satisfies the identityfor difference quotients.

When the space H is a space H(W ), multiplication by W (z) acts as a partially isomet-ric transformation of C(z) onto the complementary space M(W ) to the space H(W ) inC(z). Multiplication by U(z) acts as a partially isometric transformation of C(z) onto thecomplementary spaceM(U) to the space H(U) in C(z). Since the space H(U) is containedcontractively in the space H(W ), the space M(W ) is contained contractively in the spaceM(U). A contractive transformation of C(z) into itself is defined by taking f(z) into thesolution g(z) of the equation

U(z)g(z) = W (z)f(z)

which is orthogonal to the kernel of multiplication by U(z). The transformation takeszf(z) into zg(z) whenever it takes f(z) into g(z) since the kernel of multiplication by U(z)contains [h(z) − h(0)]/z whenever it contains h(z). A power series V (z) with operator


coefficients exist such that multiplication by V (z) is the contractive transformation of C(z)into itself taking f(z) into g(z). The space H(V ) is orthogonal in C(z) to the kernel ofmultiplication by U(z). Multiplication by U(z) is an isometric transformation of the spaceH(V ) onto the orthogonal complement in the space H(W ) of the space H(U). The identity

W (z) = U(z)V (z)

is satisfied.

The factorization admits a reformulation using Herglotz spaces.

Theorem 2. If W (z) is a power series with operator coefficients such that multiplicationby W (z) is a contractive transformation of C(z) into itself and if the adjoint of multipli-cation by W (z) acts as a partially isometric transformation of C(z) onto a Herglotz spaceL(φ), then the orthogonal complement in the space H(W ) of elements W (z)f(z) with f(z)a polynomial element of the Herglotz space L(1 − φ) is a Hilbert space H(U) which iscontained isometrically in the space H(W ) and which satisfies the identity for differencequotients. A power series V (z) with operator coefficients exists such that multiplication byV (z) is a contractive transformation of C(z) into itself whose range is orthogonal to thekernel of multiplication by U(z) and which satisfies the identity

W (z) = U(z)V (z).

Multiplication by U(z) is an isometric transformation of the space H(V ) onto the orthog-onal complement of the space H(U) in the space H(W ). The adjoint of multiplication byV (z) acts as a partially isometric transformation of C(z) onto a Herglotz space L(ψ) suchthat the Herglotz space L(1−ψ) is contained isometrically in the Herglotz space L(1−φ) andcontains the polynomial elements of L(1− φ). The adjoint of multiplication by U(z) actsas a partially isometric transformation of C(z) onto a Herglotz space L(θ). The adjoint ofmultiplication by V (z) acts as an isometric transformation of the Herglotz space L(1− θ)onto the Herglotz space L(ψ−φ) which is the orthogonal complement in the Herglotz spaceL(1 − ψ) in the Herglotz space L(1− φ). A dense set of elements of the space H(V ) areproducts V (z)f(z) with f(z) a polynomial element of the space L(1− φ).

Proof of Theorem 2. The construction of the space H(U) is an equivalent constructionof the greatest subspace of the space H(W ) which is contained isometrically in the spaceH(W ) and which satisfies the identity for difference quotients. The construction of thespace H(V ) repeats a construction which has previously been made with a statement ofconsequences for associated Herglotz spaces. Proofs are an application of complementation.Care is taken to avoid the conclusion that the polynomial elements of the Herglotz spaceL(1− ψ) are dense in the space. Density is asserted only for images of polynomials in thespace H(V ).


If a Herglotz space L(φ) is contained contractively in C(z) and if the polynomial elementsof the space are dense in the space, a partially isometric transformation of C(z) onto the


Herglotz space L(φ) exists which takes [f(z)−f(0)]/z into [g(z)−g(0)]/z whenever it takesf(z) into g(z). Such transformations are not unique. A power series B(z) with operatorcoefficients exists such that multiplication by B(z) is a contractive transformation of C(z)into itself whose adjoint coincides with the partially isometric transformation of C(z) ontothe Herglotz space.

A partially isometric transformation of C(z) onto the Herglotz space exists which takes[f(z) − f(0)/z into [g(z) − g(0)]/z whenever it takes f(z) into g(z) and whose kernelcontains zf(z) whenever it contains f(z). Such transformations are essentially unique. Apower series A(z) with operator coefficients exists such that multiplication by A(z) is acontractive transformation of C(z) into itself whose adjoint coincides with the partiallyisometric transformation of C(z) onto the Herglotz space.

A power series C(z) with operator coefficients exists such that multiplication by C(z)is a partially isometric transformation of C(z) into itself such that the identity

B(z) = A(z)C(z)

is satisfied.

Convex decompositions of Hilbert spaces of power series with vector coefficients whichsatisfy the inequality for difference quotients leave fixed the elements of the greatest sub-space which satisfies the identity for difference quotients. The determination of extremepoints is reduced to spaces containing no nonzero element in the greatest subspace satis-fying the identity for difference quotients.

Theorem 3. If a Hilbert space H of power series with vector coefficients which satisfiesthe inequality for difference quotients is a convex combination

H = (1− t)H+ + tH−

of Hilbert spaces H+ and H− of power series with vector coefficients which satisfy theinequality for difference quotients, then the greatest subspace H(U) of H which satisfies theidentity for difference quotients is contained isometrically in the spaces H+ and H−. Hilbertspaces H′+,H′−, and H′ which satisfy the inequality for difference quotients and which areorthogonal in C(z) to the kernel of multiplication by U(z) exist such that multiplicationby U(z) acts as an isometric transformation of H′+ onto the orthogonal complement ofH(U) in H+, of H′− onto the orthogonal complement of H(U) in H−, and of H′ onto theorthogonal complement of H(U) in H. The space

H′ = (1− t)H′+ + tH′−

is a convex combination of H′+ and H′−.

Proof of Theorem 3. If an element

h(z) = (1− t)f(z) + tg(z)


of the greatest subspace of H which satisfies the identity for difference quotients is aminimal convex combination of f(z) in H+ and g(z) in H−, then f(z) and g(z) are equalto h(z) with the norm of f(z) in H+ and the norm of g(z) in H− equal to the norm of h(z)in H. The construction of the space H′ from H applies to the construction of the space H′+from H+ and to the construction of the space H′− from H−. The convex decomposition ofthe space H′ restates the convex decomposition of the space H.


The extension space of a Herglotz space L(φ) is a Hilbert space E(φ) of Laurent serieswith vector coefficients such that multiplication by z is an isometric transformation of thespace onto itself and such that a partially isometric transformation of the space onto thespace L(φ) is defined by taking a Laurent series into the power series which has the samecoefficient of zn for every nonnegative integer n.

The extension space of the Herglotz space C(z) of square summable power series is thespace ext C(z) of square summable Laurent series

f(z) =∑

anzn

with‖f(z)‖2 =

∑|an|2.

If E(φ) and E(ψ) are extension spaces of Herglotz spaces L(φ) and L(ψ), then a Herglotzspace

L(φ+ ψ) = L(φ) ∨ L(ψ)

exists in which the Herglotz spaces L(φ) and L(ψ) are contained contractively as com-plementary spaces. The extension spaces E(φ) and E(ψ) are contained contractively ascomplementary spaces in the extension space

E(φ+ ψ) = E(φ) ∨ E(ψ).

A Herglotz space L(θ) exists such that the extension space

E(θ) = E(φ) ∧ E(ψ)

is the intersection space of the extension spaces E(φ) and E(ψ).

A Herglotz space L(φ− θ) exists such that the extension space E(φ− θ) is the comple-mentary space in the extension space E(φ) of the extension space E(θ). A Herglotz spaceL(ψ − θ) exists such that the extension space E(ψ − θ) is the complementary space inthe extension space E(ψ) of the extension space E(θ). A Herglotz space L(φ + θ) existssuch that the extension space E(φ+ θ) is the complementary space in the extension spaceE(φ+ ψ) of the extension space E(ψ− θ). A Herglotz space L(ψ + θ) exists such that theextension space E(ψ + θ) is the complementary space in the extension space E(φ + ψ) ofthe extension space E(φ− θ). The extension space

E(φ) = 12E(φ+ θ) + 1

2E(φ− θ)


is a convex combination of the extension spaces E(φ + θ) and E(φ − θ). The extensionspace

E(ψ) = 12 E(ψ + θ) + 1

2E(ψ − θ)

is a convex combination of the extension spaces E(ψ + θ) and E(ψ − θ).A partially isometric transformation of the extension space E(φ+ ψ) onto the Herglotz

space L(φ + ψ) is defined by taking a Laurent series into the power series which has thesame coefficient of zn for every nonnegative integer n. The restriction acts as a partiallyisometric transformation of the extension space E(φ) onto the Herglotz space L(φ), of theextension space E(ψ) onto the Herglotz space L(ψ), and of the extension space E(θ) ontothe Herglotz space L(θ). Partially isometric transformations are obtained of the extensionspaces E(φ+ θ) and E(φ− θ) onto the Herglotz spaces L(φ+ θ) and L(φ− θ) and of theextension spaces E(ψ + θ) and E(ψ − θ) onto the Herglotz spaces L(ψ + θ) and L(ψ − θ).

The Herglotz space L(θ) is contained contractively in the Herglotz spaces L(φ) andL(ψ). The Herglotz space L(φ−θ) is the complementary space in the Herglotz space L(φ)of the Herglotz space L(θ). The Herglotz space L(ψ − θ) is the complementary space inthe Herglotz space L(ψ) of the Herglotz space L(θ). The Herglotz space L(φ + θ) is thecomplementary space in the Herglotz space L(φ+ψ) of the Herglotz space L(ψ− θ). TheHerglotz space L(ψ+ θ) is the complementary space in the Herglotz space L(φ+ψ) of theHerglotz space L(φ− θ). The Herglotz space

L(φ) = 12L(φ+ θ) + 1

2L(φ− θ)

is a convex combination of the Herglotz spaces L(φ+ θ) and L(φ− θ). The Herglotz space

L(ψ) = 12L(ψ + θ) + 1

2L(ψ − θ)

is a convex combination of the Herglotz spaces L(ψ + θ) and L(ψ − θ).

An application of the convex decomposition is the determination of the extreme pointsof the convex set of Herglotz spaces which are contained contractively in the space C(z) ofsquare summable power series. A Herglotz space L(φ), which belongs to the convex set,is an extreme point of the convex set if, and only if, the extension space E(φ) is containedisometrically in the space ext C(z) of square summable Laurent series.

An existence theorem for invariant subspaces for isometric transformations of a Hilbertspace onto itself is an application of extension spaces of Herglotz spaces. Such a transfor-mation is unitarily equivalent to multiplication by z in the extension space E(φ) of someHerglotz space L(φ). A decomposition

φ(z) = φ+(z) + φ−(z)

holds for Herglotz spaces L(φ+) and L(φ−) such that the operator valued function rep-resented by φ+(z) admits an analytic extension to the upper half–plane and the operatorvalued function represented by φ−(z) admits an analytic extension to the lower half–plane.


The intersection space E(φ+) ∧ E(φ−) is an extension space whose elements are sums

a(z − 1)−1 + b(z + 1)−1

for vectors a and b. The decomposition can be made so that the intersection space containsno nonzero element. This can be done for example so that the space E(φ−) contains everyelement

a(z + 1)−1

of the space E(φ) and the space E(φ+) contains every element

b(z − 1)−1

of the space E(φ). The spaces E(φ+) and E(φ−) are then contained isometrically in thespace E(φ) and are invariant subspaces of every continuous transformation of the spaceE(φ) into itself which commute with multiplication by z.

The decomposition of Herglotz functions is an adaptation to the unit disk of the Poissonrepresentation of functions which are analytic and have nonnegative real part in the upperhalf–plane. The transition from complex valued functions to operator valued functionscauses no difficulty in the Poisson representation.

The original proof of existence of invariant subspaces for isometric transformations of aHilbert space onto itself was presented by Hilbert in lectures at the Gottingen mathematicalinstitute. Although the lecture notes taken by his student Ernst Hellinger have been lost,his subsequent research indicates a formulation of the existence theorem related to Herglotzspaces. These spaces are named after a member of the Gottingen institute who was clearlyinfluenced by Hilbert.

The extension space of a Hilbert space H of power series with vector coefficients whichsatisfies the identity for difference quotients is a Hilbert space ext H which is containedcontractively in ext C(z) and which contains the space H isometrically. The orthogonalcomplement of H in ext H is contained isometrically in ext C(z) and coincides with theorthogonal complement of C(z) in ext C(z). The complementary space to ext H in ext C(z)is isometrically equal to the complementary space to H in C(z). When there is no nonzeroelement in the greatest subspace of H satisfying the identity for difference quotients, theorthogonal complement of H in ext H is the set of elements of ext H on which the inclusionin ext C(z) is isometric.

A convex decomposition applies to a Hilbert space of power series with vector coeffi-cients which satisfies the inequality for difference quotients when the identity for differencequotients is not satisfied.

Theorem 4. An extreme point of the convex set of Hilbert spaces of power series withvector coefficients which satisfy the inequality for difference quotients is a space whichsatisfies the identity for difference quotients.

Proof of Theorem 4. A convex decomposition is made of a Hilbert space H of powerseries with vector coefficients which satisfies the inequality for difference quotients. The


decomposition is nontrivial when the space does not satisfy the identity for differencequotients. It can be assumed without loss of generality that the greatest subspace of Hwhich satisfies the identity for difference quotients contains no nonzero element.

Division by z is a contractive transformation of ext H into itself which acts as anisometric transformation of ext H onto a Hilbert space which is contained contractively inext H and whose complementary space in ext H is a Hilbert space B which is containedcontractively in ext H. The intersection space of B and its complementary space in extH is a Hilbert space ∆ which is contained contractively in B and in the complementaryspace to B in ext H. The complementary space to ∆ in B is a Hilbert space B+ which iscontained contractively in B. A Hilbert space B−, which is contained contractively in extC(z) is defined by its complementary space in ext C(z), which is equal to the complementaryspace to ∆ in the space which is complementary to B in ext C(z). The space B is containedcontractively in the space B−. The space ∆ is isometrically equal to the complementaryspace to B in B−. The convex decomposition

B = (1− t)B− + tB+

applies with t and 1− t equal.

Multiplication by zn acts as an isometric transformation of B onto a Hilbert space Bnwhich is contained contractively in ext H for every nonpositive integer n. If elements cnof Bn are chosen for nonpositive integers n with convergent sum∑

‖cn‖2Bn ,

then the sumc =

∑cn

converges in the metric topology of ext H and represents an element c which satisfies theinequality

‖c‖2ext H ≤∑‖cn‖2Bn .

Every element c of ext H admits a representation for which equality holds.

Multiplication by zn acts as an isometric transformation of B+ onto a Hilbert space Bn+which is contained contractively in ext H for every nonpositive integer n. If elements anof Bn+ are chosen for nonpositive integers n with convergent sum∑

‖an‖2Bn+

then the suma =

∑an

converges in the metric topology of ext H and represents an element a which satisfies theinequality

‖a‖2ext H ≤∑‖an‖2Bn

+.


The represented elements of ext H form a Hilbert space ext H−, which is contained con-tractively in ext H, such that the inequality

‖a‖2ext H+≤∑‖an‖2Bn

+

is always satisfied and such that equality holds in some representation. Division by z is acontractive transformation of ext H+ into itself.

Multiplication by zn acts as an isometric transformation of B− onto a Hilbert space Bn−which is contained contractively in ext C(z) for every nonpositive integer n. If element bnof Bn− are chosen for nonpositive integers n with convergent sum∑

‖bn‖2Bn− ,

then the sumb =

∑bn

converges in the metric topology of ext C(z) and represents an element b which satisfiesthe inequality

‖b‖2ext C(z) ≤∑‖bn‖2Bn− .

The represented elements of ext C(z) form a Hilbert space ext H−, which is containedcontractively in ext C(z), such that the inequality

‖b‖2ext H− ≤∑‖bn‖2Bn−

is always satisfied and such that equality holds in some representation. Division by z is acontractive transformation of ext H− into itself.

The convex combination

ext H = (1− t) ext H+ + t ext H−

applies with t and 1− t equal. The elements of ext H on which the inclusion of ext H inext C(z) is isometric are elements of ext H+ on which the inclusion of ext H+ in ext C(z)is isometric and are elements of ext H− on which the inclusion of ext H− in ext C(z) isisometric. The orthogonal complement of these elements in ext H+ is a Hilbert spaceH+ of power series with vector coefficients which is contained isometrically in ext H+ andwhich satisfies the inequality for difference quotients. The orthogonal complement of theseelements in ext H− is a Hilbert space H− of power series with vector coefficients which iscontained isometrically in the space ext H− and which satisfies the inequality for differencequotients. The convex combination

H = (1− t)H+ + tH−

applies with t and 1− t equal.


When the space H does not satisfy the identity for difference quotients, the space∆ contains a nonzero element, the spaces B+ and B− are not isometrically equal, thespaces ext H+ and ext H− are not isometrically equal, and the spaces H+ and H− are notisometrically equal.


If W (z) is a power series with operator coefficients such that multiplication by W (z) isa contractive transformation of C(z) into itself, then multiplication by W (z) has a uniqueextension as a contractive transformation of ext C(z) into itself which takes f(z) into g(z)whenever it takes zf(z) into zg(z). Multiplication by W (z) is a contractive transformationof ext C(z) into itself which commutes with multiplication by z. The conjugate power series

W ∗(z) = W−0 +W−1 z +W−2 z2 + . . .

is defined with operator coefficients which are adjoints of the operator coefficients of thepower series

W (z) = W0 +W1z +W2z2 + . . .

Multiplication by W ∗(z) is a contractive transformation of C(z) into itself which extendsto a contractive transformation of the ext C(z) into itself. The adjoint of multiplication byW (z) as a transformation of ext C(z) into itself takes f(z) into g(z) when multiplicationby W ∗(z) takes z−1f(z−1) into z−1g(z−1). If the adjoint of multiplication by W (z) asa transformation of ext C(z) into itself takes f(z) into g(z) and if the coefficient of zn inf(z) vanishes for every nonnegative integer n, then the coefficient of zn in g(z) vanishes forevery nonnegative integer n. The adjoint of multiplication by W (z) as a transformation ofC(z) into itself takes an element f(z) of C(z) into the power series g(z) which has the samecoefficient of zn for every nonnegative integer n as the Laurent series obtained from f(z)under the adjoint of multiplication by W (z) as a transformation of ext C(z) into itself.

Invariant subspaces for contractive transformations of a Hilbert space into itself areconstructed by factorization of transfer functions in canonical models. If A(z) and B(z) arepower series with operator coefficients such that multiplication by A(z) and multiplicationby B(z) are contractive transformations of C(z) into itself and if

B(z) = A(z)C(z)

for a power series C(z) with operator coefficients such that multiplication by C(z) is acontractive transformation of C(z) into itself, then the space H(A) is contained contrac-tively in the space H(B). Multiplication by A(z) is a contractive transformation of thespace H(C) onto the complementary space to the space H(A) in the space H(B).

A converse result [4] applies when the identity for difference quotients is satisfied.

Theorem 5. If a space H(A) is contained contractively in a space H(B), then the equation

B(z) = A(z)C(z)


has as solution a power series C(z) with operator coefficients such that multiplicationby C(z) is a contractive transformation of C(z) into itself which annihilates elements ofthe coefficient space annihilated on multiplication by B(z) and whose adjoint annihilateselements of the coefficient space annihilated on multiplication by A(z).

Proof of Theorem 5. If r is a nonnegative integer, multiplication by the power series

Ar(z) = zrA(z)

and by the power seriesBr(z) = zrB(z)

is a contractive transformation of C(z) into itself. The set of polynomial elements of C(z)of degree less than r is a Hilbert space which is contained isometrically in C(z) and whichis contained isometrically in the spaces H(Ar) and H(Br). Multiplication by zr acts as anisometric transformation of the space H(A) onto the orthogonal complement in the spaceH(Ar) of the polynomial elements of degree less than r and of the space H(B) onto theorthogonal complement in the space H(Br) of the polynomial elements of degree less thanr.

An isometric transformation of the space H(Ar) into the space ext H(A) is defined bytaking f(z) into z−rf(z) for every nonnegative integer r. The union of the images of thespaces H(Ar) is dense in the space ext H(A).

An isometric transformation of the space H(Br) into the space ext H(B) is defined bytaking g(z) into z−rg(z) for every nonnegative integer r. The union of the images of thespaces H(Br) is dense in the space ext H(B).

The space ext H(A) is contained contractively in the space ext H(B) since the spaceH(Ar) is contained contractively in the space H(Br) for every nonnegative integer r.

The space ext H(A) is contained contractively in ext C(z). Multiplication by A(z) actsas a partially isometric transformation of ext C(z) onto the complementary space to thespace ext H(A) in ext C(z). An element h(z) of ext C(z) admits a minimal decomposition

h(z) = h(z)−A(z)f(z) + A(z)f(z)

withh(z)−A(z)f(z)

in ext H(A) and f(z) in ext C(z). Equality holds in the inequality

‖h(z)‖2 ≤ ‖h(z)− A(z)f(z)‖2ext H(A) + ‖f(z)‖2.

The element f(z) of ext C(z) is obtained from h(z) on multiplication by A∗(z−1).

The space ext H(B) is contained contractively in ext C(z). Multiplication by B(z) actsas a partially isometric transformation of ext C(z) onto the complementary space to thespace ext H(B) in ext C(z). An element h(z) of ext C(z) admits a minimal decomposition

h(z) = h(z)−B(z)g(z) +B(z)g(z)


withh(z)−B(z)g(z)

in ext H(B) and g(z) in ext C(z). Equality holds in the inequality

‖h(z)‖2 ≤ ‖h(z)−B(z)g(z)‖2ext H(B) + ‖g(z)‖2.

The element g(z) of ext C(z) is obtained from h(z) under multiplication by B∗(z−1).

Since the elementh(z)−A(z)f(z)

of the space ext H(A) is obtained from h(z) under the adjoint of the inclusion of the spacein ext C(z), since the element

h(z)−B(z)g(z)

of the space ext H(B) is obtained from h(z) under the adjoint of the inclusion of the spacein ext C(z), and since the space ext H(A) is contained contractively in the space ext H(B),the element

h(z)−A(z)f(z)

of the space ext H(A) is obtained from the element

h(z)−B(z)g(z)

of the space ext H(B) under the adjoint of the inclusion of the space ext H(A) in the spaceext H(B). The inequality

‖h(z)− A(z)f(z)‖ext H(A) ≤ ‖h(z)−B(z)g(z)‖ext H(B)

holds since the adjoint of a contractive transformation is contractive. This completes theproof of the inequality

‖g(z)‖ ≤ ‖f(z)‖.

The contractive transformation taking f(z) into g(z) admits a continuous extensionas a contractive transformation of the orthogonal complement in ext C(z) of the kernelof multiplication by A(z) into the orthogonal complement in ext C(z) of the kernel ofmultiplication by B(z). The domain of the transformation contains z−1f(z) whenever itcontains f(z). The transformation takes z−1f(z) into z−1g(z) whenever it takes f(z) intog(z).

The transformation has a contractive extension which takes zf(z) into zg(z) wheneverit takes f(z) into g(z). The extension is unique as a transformation of the orthogonalcomplement in ext C(z) of the elements whose coefficients are in the kernel of multiplicationby A(z) into the orthogonal complement in ext C(z) of the elements whose coefficients arein the kernel of multiplication by B(z).

A power series C(z) with operator coefficients exists such that multiplication by C(z)is a contractive transformation of C(z) into itself and such that multiplication by C∗(z−1)


extends the contractive transformation of the orthogonal complement in ext C(z) of theelements whose coefficients lie in the kernel of multiplication by A(z) into the orthogonalcomplement in ext C(z) of the elements whose coefficients lie in the kernel of multiplicationby B(z). The power series is unique when the operator coefficients of C(z) annihilateelements of the coefficient space in the kernel of multiplication by B(z) and the adjointsof operator coefficients of C(z) annihilate elements of the coefficient space in the kernel ofmultiplication by A(z).

The factorizationB(z) = A(z)C(z)

follows from the identityB∗(z−1) = C∗(z−1)A∗(z−1).


A contractive inclusion of a space H(A) in a space H(B) is an example of a transfor-mation which takes [f(z) − f(0)]/z into [g(z)− g(0)]/z whenever it takes f(z) into g(z).When

A(z) = D(z)B(z)

with D(z) a power series with operator coefficients such that multiplication by D(z) is acontractive transformation of C(z) into itself, the adjoint of multiplication by D(z) as atransformation of C(z) into itself acts as a contractive transformation of the space H(A)into the space H(B) which takes [f(z)−f(0)]/z into [g(z)−g(0)]/z whenever it takes f(z)into g(z).

An application of commutant lifting [4] is made to the structure of contractive transfor-mations of a space H(A) into a space H(B) which take [f(z)− f(0)]/z into [g(z)− g(0)]/zwhenever they take f(z) into g(z).

Theorem 6. If a contractive transformation T of a spaceH(A), which satisfies the identityfor difference quotients, into a space H(B) takes [f(z)−f(0)]z into [g(z)−g(0)]/z wheneverit takes f(z) into g(z), then a contractive transformation T ′ of the space H(A′),

A′(z) = zA(z),

into the space H(B′),B′(z) = zB(z),

exists which extends T and which takes [f(z) − f(0)]/z into [g(z) − g(0)]/z whenever ittakes f(z) into g(z).

Proof of Theorem 6. The construction of T ′ is an application of complementation. Thecoefficient space C is contained isometrically in the spaces H(A′) andH(B′). Multiplicationby z is an isometric transformation of the space H(A) into the orthogonal complement ofC in the space H(A′) and of the space H(B) onto the orthogonal complement of C inthe space H(B′). Since T ′ is defined to agree with T on the space H(A), it remains to


define T ′ on the complementary space to the space H(A) in the space H(A′), which is anorthogonal complement since the space H(A) satisfies the identity for difference quotientsby hypothesis.

A contractive transformation of the complementary space to the space H(A) in thespace H(A′) into the orthogonal complement of C in the space H(B′) is defined by takingf(z) into g(z) when T takes [f(z)−f(0)]/z into [g(z)−g(0)]/z with g(0) equal to zero. Thetransformation acts as a partially isometric transformation of the complementary space tothe space H(A) in the space H(A′) onto a Hilbert space P which is contained contractivelyin the orthogonal complement of C in the space H(B′).

A contractive transformation of the space H(A) into the orthogonal complement of Cin the space H(B′) is defined by taking f(z) into g(z) when T takes [f(z)− f(0)]/z into[g(z) − g(0)]/z with g(0) equal to zero. The transformation acts as a partially isometrictransformation of the space H(A) onto a Hilbert space Q which is contained contractivelyin the orthogonal complement of C in the space H(B′).

A Hilbert space P ∨ Q exists which is contained contractively in the orthogonal com-plement of C in the space H(B′) and which contains the spaces P and Q contractively ascomplementary spaces. A contractive transformation of the space H(A′) into the orthogo-nal complement of C in the space H(B′) is defined by taking f(z) into g(z) when T takes[f(z)− f(0)]/z into [g(z)− g(0)]/z with g(0) equal to zero. The transformation acts as apartially isometric transformation of the space H(A′) onto the space P ∨ Q.

A Hilbert space M is defined as the set of elements f(z) of the space H(B′) such thatf(z)− f(0) belongs to P ∧ Q with scalar product determined by the identity

‖f(z)‖2M = ‖f(z)− f(0)‖2P∨Q + |f(0)|2.

The space M is contained contractively in the space H(B′).

Since T is a contractive transformation of the space H(A) into the space H(B), T actsas a partially isometric transformation of the space H(A) onto a Hilbert space Q′ which iscontained contractively in the space H(B). Since the space Q′ is contained contractively inthe spaceM, the complementary space to Q′ inM is a Hilbert space P ′ which is containedcontractively in M.

A partially isometric transformation of the spaceM onto the space P ∨Q is defined bytaking f(z)− f(0). Since a partially isometric transformation of Q′ onto Q is defined bytaking f(z) into f(z)− f(0), a partially isometric transformation of P ′ onto P is definedby taking f(z) into f(z)− f(0).

The transformation T ′ is defined to take an element f(z) of the complementary space tothe space H(A) in the space H(A′) into the element g(z) of the space P ′ of least norm suchthat T takes [f(z) − f(0)]/z into [g(z) − g(0)]/z. Since the complementary space to thespace H(A) is an orthogonal complement, the transformation T ′ admits a unique linearextension to the space H(A′). The transformation T ′ is contractive as a transformationof the space H(A′) into the space H(B′). The transformation T ′ extends T and takes[f(z)− f(0)]/z into [g(z)− g(0)]/z whenever it takes f(z) into g(z).



A contractive transformation of a space H(A) into a space H(B) taking [f(z)− f(0)]/zinto [g(z)− g(0)]/z whenever it takes f(z) into g(z) is the composition of an inclusion andthe adjoint of multiplication by a power series with operator coefficients when the spaceH(A) satisfies the identity for difference quotients.

Theorem 7. If a contractive transformation of a space H(A) which satisfies the identityfor difference quotients into a space H(B) takes [f(z)− f(0)]/z into [g(z)− g(0)]/z when-ever it takes f(z) into g(z), then power series C(z) and D(z) with operator coefficientsexist which define contractive multiplications to C(z) into itself such that the given trans-formation coincides on the space H(A) with the adjoint of multiplication by D(z) and suchthat the identity

A(z)C(z) = D(z)B(z)

is satisfied.

Proof of Theorem 7. Since the space H(A) satisfies the identity for difference quotientsby hypothesis, the space H(Ar) satisfies the identity for difference quotients for everynonnegative integer r. A contractive transformation Tr of the space H(Ar) into the spaceH(Br) which takes [f(z) − f(0)]/z into [g(z) − g(0)]/z whenever it takes f(z) into g(z)is constructed inductively for every nonnegative integer r. The transformation T0 is thegiven transformation of the space H(A) into the space H(B). The transformation Tr+1 isconstructed from the transformation Tr by commutant lifting so as to agree with Tr onthe space H(Ar).

A transformation of the union of the spaces H(Ar) into the union of the spaces H(Br)is defined which agrees with Tr on the space H(Ar) for every nonnegative integer r. Thetransformation takes [f(z)− f(0)]/z into [g(z)− g(0)]/z whenever it takes f(z) into g(z).Since the union of the spaces H(Ar) is dense in C(z), the transformation admits a uniquecontinuous extension as a contractive transformation of C(z) into itself. Since the extensiontakes [f(z)− f(0)]/z into [g(z)− g(0)]/z whenever it takes f(z) into g(z), it is the adjointof multiplication by D(z) for a power series D(z) with operator coefficients such thatmultiplication by D(z) is a contractive transformation of C(z) into itself.

The given transformation of the space H(A) into the space H(B) coincides on the spaceH(A) with the adjoint of multiplication by D(z). A space H(D) exists. Since the spaceH(D) is contained contractively in the space H(DB), multiplication by D(z) acts as acontractive transformation of the space H(B) onto the complementary space to the spaceH(D) in the space H(DB). An element h(z) of the space H(DB) admits a minimaldecomposition

h(z) = h(z)−D(z)g(z) +D(z)g(z)

withh(z)−D(z)g(z)

in the space H(D) and g(z) in the space H(B). Equality holds in the inequality

‖h(z)‖2H(DB) ≤ ‖h(z)−D(z)g(z)‖2H(DB) + ‖g(z)‖2.


The element g(z) of the space H(B) is obtained from h(z) under the adjoint of multipli-cation by g(z).

Since the adjoint of multiplication by D(z) takes an element h(z) of the space H(A)into an element f(z) of the space H(B), the space H(A) is contained in the space H(DB).Equality holds in the inequality

‖h(z)‖2H(DB) ≤ ‖h(z)−D(z)f(z)‖2H(D) + ‖f(z)‖2H(B).

Since the inequality

‖f(z)‖H(B) ≤ ‖h(z)‖H(A)

is satisfied, the space H(A) is contained contractively in the space H(DB). The identity

A(z)C(z) = D(z)B(z)

holds with C(z) a power series with operator coefficients such that multiplication by C(z)is a contractive transformation of C(z) into itself.


The relationship between factorization and invariant subspaces is applied in the con-struction of invariant subspaces. Nontrivial factorizations exist when the existence ofinvariant subspaces is not given by the Hilbert theorem.

Theorem 8. If the transformation taking f(z) into [f(z)− f(0)]/z of a space H(B) intoitself is not a scalar multiple of an isometric transformation of the space onto itself, thena Hilbert space H exists, which satisfies the inequality for difference quotients, which iscontained contractively in the spaceH(B) and which is not a convex combination of the leastand the greatest subspace of the space H(B), such that every contractive transformationof the space H(B) into itself, which takes [f(z)− f(0)]/z into [g(z)− g(0)]/z whenever ittakes f(z) into g(z), acts as a contractive transformation of the space H into itself.

Proof of Theorem 8. The spaceH is the range of the transformation taking f(z) into [f(z)−f(0)]/z of the space H(B) into itself. The scalar product of the space H is determined bythe identity

‖g(z)‖2H = ‖g(z)‖2H(B) + inf ‖f(z)‖2H(B)

with the greatest lower bound taken over the elements f(z) of the space H(B) such that

g(z) = [f(z)− f(0)]/z.

The space H satisfies the identity for difference quotients since the space H(B) satisfiesthe identity for difference quotients. A contractive transformation of the space H(B) intoitself which takes [f(z)− f(0)]/z into [g(z)− g(0)]/z whenever it takes f(z) into g(z) actsas a contractive transformation of the space H into itself.


The spaceH contains a nonzero element since the transformation taking f(z) into [f(z)−f(0)]/z of the space H(B) into itself is not a scalar multiple of the identity transformation.The space H is not a convex combination of the least subspace and the greatest subspaceof the space H(B) when the range of the transformation does not contain every elementof the space H(B) or when the kernel of the transformation contains a nonzero element.When the range of the transformation contains every element of the space H(B) and thekernel of the transformation contains no nonzero element, the space H is not a convexcombination of the least subspace and the greatest subspace of the space H(B) since thetransformation is not a constant multiple of an isometric transformation of the spaceH(B)onto itself.


An invariant subspace is constructed from a factorization when the identity for differencequotients is satisfied. Spaces which satisfy the identity for difference quotients are theextreme points of a compact convex set of spaces which satisfy the inequality for differencequotients. The compact convex set is the closed convex span of its extreme points by theKrein–Milman theorem.

Theorem 9. If a space H(B) satisfies the identity for difference quotients and if thetransformation of the space into itself which takes f(z) into [f(z)− f(0)]/z is not a scalarmultiple of an isometric transformation, then a space H(A), which satisfies the identityfor difference quotients, which is contained contractively in the space H(B), and which isneither the least nor the greatest subspace, exists such that every contractive transformationof the space H(B) into itself, which takes [f(z)− f(0)]/z into [g(z)− g(0)]/z whenever ittakes f(z) into g(z), acts as a contractive transformation of the space H(A) into itself.

Proof of Theorem 9. The convex set consists of the Hilbert spaces, which are containedcontractively in the space H(B) and which satisfy the inequality for difference quotients,such that every contractive transformation of the space H(B) into itself, which takes[f(z)− f(0)]/z into [g(z)− g(0)]/z whenever it takes f(z) into g(z), acts as a contractivetransformation of the subspace into itself.

The convex set of all Hilbert spaces which are contained contractively in C(z) is compact.Compactness of the convex set of spaces which satisfy the inequality for difference quotientsis verified by showing that the set is closed since such spaces are contained contractively inC(z). If J is the inclusion of the space in C(z) and if J∗ is the adjoint transformation of C(z)into the space, then the nonnegative self–adjoint transformation JJ∗ is contractive whenthe space is contained contractively in C(z). If S is multiplication by z as a transformationof C(z) into itself and if S∗ is the adjoint transformation taking f(z) into [f(z)− f(0)]/z,then the inequality

S∗(1− JJ∗)S ≤ 1− JJ∗

is equivalent to the inequality for difference quotients since it declares the adjoint of mul-tiplication by z a contractive transformation of the complementary space into itself. Theidentity implies that the convex set is compact.


A compact convex set is the set of all Hilbert spaces which are contained contractivelyin the space H(B) and which satisfy the inequality for difference quotients such that everycontractive transformation of the space H(B) into itself which takes [f(z)− f(0)]/z into[g(z)− g(0)]/z whenever it takes f(z) into g(z) acts as a contractive transformation of theHilbert space into itself. If T is the transformation and if J is the inclusion of the Hilbertspace in the space H(B), the constraint place on the Hilbert space is equivalent to theinequality

|〈TJJ∗f, JJ∗g〉H(B)|2 ≤ 〈JJ∗f, f〉H(B)〈JJ∗g, g〉H(B)

for all elements f(z) and g(z) of the space H(B). The convex set is compact since it isclosed.

A Hilbert spaceH = (1− t)H+ + tH−

of power series with vector coefficients which satisfies the inequality for difference quotientsis a convex combination of Hilbert spaces, with t and 1 − t equal, of Hilbert spaces H+

and H− of power series with vector coefficients which satisfy the inequality for differencequotients. The spaces H+ and H− are not isometrically equal when the space H does notsatisfy the identity for difference quotients.

The spaceH is dense in the space H−, which is contained contractively in the augmentedspace H′. The elements of H′ are the elements f(z) of C(z) such that [f(z) − f(0)]/zbelongs to H. The scalar product in the augmented space is determined by the identityfor difference quotients

‖[f(z)− f(0)]/z‖2H = ‖f(z)‖2H′ − |f(0)|2.

The spaces H+ and H− are contained contractively in the space H(B) when the space His contained contractively in the space H(B) since the space H(B) satisfies the identity fordifference quotients by hypothesis.

The extension spaceext H = (1− t)ext H+ + t ext H−

is a convex combination, with t and 1−t equal, of the extension space ext H+ and ext H−.Division by z acts as a contractive transformation of the space ext H into itself, of thespace ext H+ into itself, and of the space ext H− into itself. Each transformation actsas an isometric transformation of the domain Hilbert space onto a Hilbert space which iscontained contractively in the domain Hilbert space.

The convex decompositionB = (1− t)B+ + tB−

applies with B the complementary space to the range of division by z in the space ext H,B+ the complementary space to the range of division by z in the space H+, and B− thecomplementary space to the range of division by z in the space ext H−.

Division by z acts as an isometric transformation of the space ext H onto the comple-mentary space to B in ext H. The intersection space of B and its complementary space in


ext H is a Hilbert space ∆ which is contained contractively in B and in its complementaryspace in ext H. The space B+ is the complementary space to ∆ in B. The space ∆ iscontained contractively in the space B−. The space B is the complementary space to ∆ inB−.

A contractive transformation of the space H(B) into itself which takes [f(z)− f(0)]/zinto [g(z)− g(0)]/z whenever it takes f(z) to g(z) is the restriction of a contractive trans-formation of C(z) into itself which takes [f(z) − f(0)]/z into [g(z) − g(0)]/z whenever ittakes f(z) into g(z) since the space H(B) satisfies the identity for difference quotients byhypothesis. A power series W (z) with operator coefficients exists such that multiplicationby W (z) is a contractive transformation of C(z) into itself whose adjoint extends the trans-formation of the space H(B) into itself. Multiplication by W ∗(z−1) acts as a contractivetransformation of the space ext H(B) into itself. If the adjoint of multiplication by W (z)as a transformation of C(z) into itself acts as a contractive transformation of the spaceH into itself, then multiplication by W ∗(z−1) acts as a contractive transformation of thespace ext H into itself.

The extension space of the Hilbert space B(z) of square summable power series withcoefficients in B is the Hilbert space ext B(z) of square summable Laurent series withcoefficients in B. The set of elements of ext B(z) whose coefficient of zn vanishes when nis positive is a Hilbert space B(z−1) which is contained isometrically in ext B(z). If∑

hnzn

is an element of B(z−1), the series ∑znhn(z)

converges to an element of ext H which satisfies the inequality

‖∑

znhn(z)‖2ext H ≤∑‖hn(z)‖2B.

Every element of ext H admits a representation as∑znhn(z)

for which equality holds.

The Hilbert space B+(z−1) of square summable power series with coefficients in B+ andvanishing coefficient of zn for positive n is applied to the structure of the space ext H+. If∑

fnzn

is an element of B+(z−1), the series ∑znfn(z)


converges to an element of ext H+ which satisfies the inequality

‖∑

znfn(z)‖2ext H+≤∑‖fn(z)‖2B+

.

Every element of ext H+ admits a representation as∑znfn(z)


The Hilbert space B−(z−1) of square summable power series with coefficients in B− andvanishing coefficient of zn for positive n is applied to the structure of the space ext H−. If∑

gnzn

is an element of B−(z−1), the series ∑zngn(z)

converges to an element of ext H− which satisfies the inequality

‖∑

zngn(z)‖2ext H− ≤∑‖gn(z)‖2B− .

Every element of ext H− admits a representation as∑zngn(z)


The space B+(z−1) is the image of a space contained isometrically in the space B(z−1)which is the orthogonal complement of the elements of B(z−1) which represent the originof ext H. The orthogonal projection of the space B(z−1) onto the space B+(z−1) takesz−1f(z) into z−1g(z) whenever it takes f(z) into g(z). The adjoint of the inclusion of thespace ext H+ in the space ext H takes

B∗(z−1)∑

znfn(z)

intoB∗(z−1)

∑zngn(z)

whenever it takes ∑znfn(z)

into ∑zngn(z).

Multiplication by B∗(z−1) is a contractive transformation of the space ext H+ into itself.Multiplication by B∗(z−1) is a contractive transformation of the space ext H− into itself.

The adjoint of multiplication by B(z) as a transformation of C(z) into itself acts as acontractive transformation of the space H+ into itself and of the space H− into itself.


The computation of extreme points completes the proof of existence of invariant sub-spaces.


Theorem 10. A continuous transformation of a Hilbert space into itself which is nota scalar multiple of the identity transformation admits a closed invariant subspace otherthan the least and the greatest subspace which is an invariant subspace for every continuoustransformation commuting with the given transformation.

Proof of Theorem 10. The invariant subspace is essentially due to Hilbert when the trans-formation is the scalar multiple of an isometric transformation. The invariant subspace canbe found by applying the present constructions in the context of Herglotz spaces. Whenthe transformation is not a scalar multiple of an isometric transformation, the transforma-tion can be assumed contractive with iterates which converge to the origin in their actionon every element of the Hilbert space. The transformation can be assumed to take f(z)into [f(z) − f(0)]/z in a space H(B) which is contained isometrically in C(z) for somecoefficient Hilbert space C.

A Hilbert space H(A), which is contained contractively in the space H(B), which sat-isfies the identity for difference quotients, which contains a nonzero element, and whichis not isometrically equal to the space H(B), such that every contractive transformationof the space H(B) into itself, which takes [f(z)− f(0)]/z into [g(z)− g(0)]/z whenever ittakes f(z) into g(z), acts as a contractive transformation of the space H(A) into itself.

A power series C(z) with operator coefficients exists such that multiplication by C(z)is a contractive transformation of C(z) into itself, such that the identity

B(z) = A(z)C(z)

is satisfied, such that multiplication by C(z) annihilates every element of the coefficientspace in the kernel of multiplication by B(z), and such that multiplication by C∗(z) anni-hilates elements of the coefficient space in the kernel of multiplication by A(z).

Since the space H(B) is contained isometrically in C(z), multiplication by B(z) is apartially isometric transformation of C(z) into itself. The kernel of multiplication by B(z)contains [f(z)− f(0)]/z whenever it contains f(z) since the orthogonal complement of thekernel contains zf(z) whenever it contains f(z).

Since the space H(A) satisfies the identity for difference quotients, multiplication byA(z) is a contractive transformation of C(z) into itself whose kernel contains [f(z)−f(0)]/zwhenever it contains f(z). Since the range of multiplication by C(z) as a transformationof C(z) into itself is orthogonal to the kernel of multiplication by A(z), multiplication byC(z) is a partially isometric transformation of C(z) into itself whose kernel is the kernel ofmultiplication by B(z). Since the space H(C) is orthogonal to the kernel of multiplicationby A(z), multiplication by A(z) acts as an isometric transformation of the space H(C)into the space H(B). The space H(A) is contained isometrically in the space H(B) as theorthogonal complement of the image of the space H(C).


References

1. A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math. 81


(1949), 239-255.

2. L. de Branges, Factorization and invariant subspaces, Jour. Math. Anal. Appl. 29 (1970), 163-200.

3. , Complementation in Krein spaces, Trans. Amer. Math. Soc. 205 (1988), 277-291.

4. , Krein spaces of analytic functions, J. Funct. Anal. 81 (1988), 219–259.

5. L. de Branges and J. Rovnyak, The existence of invariant subspaces, Bull. Amer. Math. Soc. 70 (1964),718-721.

6. , Square Summable Power Series, Holt, Rinehart & Winston, New York, 1966.

7. L. de Branges and L. Shulman, Perturbations of unitary transformations, J. Math. Anal. Appl. 23

(1968), 294–326.

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A PROOF OF THE INVARIANT SUBSPACE CONJECTURE Louis de Branges